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Stochastic Model of a Micro Agents population

Stochastic Model of a Micro Agents population. Dejan Milutinovic dejan@isr.ist.utl.pt. Outline. Motivating problem. Introduction to Math. Analysis. Mathematical Analysis. Applications. Biology. Robotics. Motivating problem. T-Cell. CD3. peptide. APC. MHC.

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Stochastic Model of a Micro Agents population

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  1. Stochastic Model ofa Micro Agents population Dejan Milutinovic dejan@isr.ist.utl.pt

  2. Outline Motivating problem Introduction to Math. Analysis Mathematical Analysis Applications Biology Robotics

  3. Motivating problem T-Cell CD3 peptide APC MHC T-Cell Receptor (TCR) triggering T-Cell, CD3 receptor, Antigen Presenting Cell (APC), peptide-MHC complex

  4. Introduction to Math. Analysis - T-Cell - APC T-Cell population

  5. Introduction to Math. Analysis q(246)=connected q(123)=never_connected - T-Cell - APC q(235)=disconnected T-Cell population Complex System !!!

  6. Introduction to Math. Analysis How the Micro Dynamics of the Individuals propagates to the Dynamics of Macro observations ?

  7. MathematicalAnalysis q=1 q=2 u(t)=a u(t)=a u(t)=b The Micro Agent model of the T-Cell q=3 1 –never connected, 2 - connected, 3- disconnected, a-connection, b-disconnection

  8. MathematicalAnalysis Initial condition (x0,q0) Input event sequence Continuous output Y(t) A u(t) u(t) Y(t) a b c Deterministic system Micro Agent (A)

  9. MathematicalAnalysis A A A Determinist. system Stoch. Stoch. process process SA Stochastic system Stochastic Micro Agent (SA)

  10. MathematicalAnalysis Micro and Macro Dynamics relation • Statistical Physics reasoning (Boltzman distribution) Dual Meaning of the State Probability Density Function • PDF function describes the state probability of one A • Looking to the large population of A, PDF is a normalized distribution of the state occupancy by all A • Micro dynamics of A and macro dynamics of A population are related through the state PDFs

  11. MathematicalAnalysis . . . Micro Agent Stochastic Execution A stochastic process (x(t),q(t))X  Q is called a Micro Agent Stochastic Execution iff a Micro Agent stochastic input event sequence e(n),nN, 0 = 0 1 2 … generates transitions such that in each interval [n,n+1), nN, q(t) q(n). xn e(n ) e(n+1 ) e(n+2 ) V V V i q 1 N f(x,N) f(x,1) f(x,i) xn-1 X x Q x1 Remark 1. The x(t) of a Stochastic Execution is a continuous time function since the transition changes only the discrete state of a Micro Agent.

  12. MathematicalAnalysis A A A Stoch. Stoch. Determinist. process process system SA Stochastic system Stochastic Micro Agent (SA) A Stochastic Micro Agent is a pair SA=(A,e(t)) where A is a Micro Agent and e(t) is a Micro Agent stochastic input event sequence such that the stochastic process (x(t),q(t))X  Q is a Micro Agent Stochastic Execution.

  13. MathematicalAnalysis q=1 q=2 u(t)=a SA u(t)=a u(t)=b q=3 Continuous Time Markov Process Micro Agent (CTMPA) A Stochastic Micro Agent is called a Continuous Time Markov Process Micro Agent iff (x(t),q(t))X  Q is a Micro Agent Continuous Time Markov Process Execution. Stochastic system

  14. MathematicalAnalysis The Continuous Time Markov Chain Micro Agent with N discrete state and state probability given by where is the probability of discrete state i and is transition rate matrix and is rateof transition from discrete state i to discrete state j. The vector of probability density functions where is probability density function of state (x,i) at time t, satisfies where is the vector of vector fields value at state (x,i).

  15. Biological application The Micro Agent model of the T-Cell q=1 q=2 u(t)=a 12 32 u(t)=a u(t)=b 23 q=3 0 –never connected, 1 - connected, 2- disconnected, a-connection, b-disconnection, ij – event rate which leads to transition from state i to state j

  16. Biological application Case I solution 12 =0.9, 23= 0 , 32 =0.5, k2 =0.5, k3=0.25

  17. Biological application Case II solution 12 =0.9, 23= 0.8 , 32 =0.9, k2 =0.5, k3=0.05

  18. Biological application Case III solution 12 =0.9, 23= 0.8 , 32 =0.9, k2 =0.5, k3=0.25

  19. Biological application

  20. Biological application

  21. Biological application

  22. Robotics application q=2 21 23 q=1 12 32 Population q=3 x1 x1 a) Source 2 Source 3 Source 1 x1 x2 b) =-/4 =0 =/4 x2 x1

  23. Robotics application

  24. Publications Milutinovic, D., Athans, M., Lima, P., Carneiro, J. “Application of Nonlinear Estimation Theory in T-Cell Receptor Triggering Model Identification”, Technical Report RT-401-02, RT-701-02, 2002, ISR/IST Lisbon, Portugal Milutinovic, D., “Stochastic Model of a Micro Agents Population”, Technical Report ISR/IST Lisbon, Portugal (working version) Milutinovic D., Lima, P., Athans, M. “Biologically Inspired Stochastic Hybrid Control of Multi-Robot Systems”, submitted to the 11th International Conference on Advanced Robotics ICAR 2003,June 30 - July 3, 2003 University of Coimbra, Portugal Milutinovic D., Carneiro J., Athans, M., Lima, P. “A Hybrid Automata Modell of TCR Triggering Dynamics” , submitted to the 11th Mediterranian Conference on Control and Automation MED 2003,June 18 - 20 , 2003, Rhodes, Greece

  25. Stochastic Model ofa Micro Agents population Dejan Milutinovic dejan@isr.ist.utl.pt

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